Seokyong Chae PhD Thesis ID 993173980

8/11/2019 Seokyong Chae PhD Thesis ID 993173980

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The Pennsylvania State University

The Graduate School

Intercollege Graduate Program in Materials

NONLINEAR FINITE ELEMENT MODELING

AND ANALYSIS OF A TRUCK TIRE

A Thesis in

Materials

by

Seokyong Chae

2006 Seokyong Chae

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

August 2006


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The thesis of Seokyong Chae was reviewed and approved* by the following:

Moustafa El-Gindy

Senior Research Associate, Applied Research Laboratory

Thesis Co-AdvisorCo-Chair of Committee

James P. Runt

Professor of Materials Science and EngineeringThesis Co-Advisor

Co-Chair of Committee

Co-Chair of the Intercollege Graduate Program in Materials

Charles E. BakisProfessor of Engineering Science and Mechanics

Ashok D. Belegundu

Professor of Mechanical Engineering

*Signatures are on file in the Graduate School.


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ABSTRACT

For an efficient full vehicle model simulation, a multi-body system (MBS) simulation is

frequently adopted. By conducting the MBS simulations, the dynamic and steady-state

responses of the sprung mass can be shortly predicted when the vehicle runs on an

irregular road surface such as step curb or pothole. A multi-body vehicle model consists

of a sprung mass, simplified tire models, and suspension system to connect them. For the

simplified tire model, a rigid ring tire model is mostly used due to its efficiency.

The rigid ring tire model consists of a rigid ring representing the tread and the belt, elastic

sidewalls, and rigid rim. Several in-plane and out-of-plane parameters need to be

determined through tire tests to represent a real pneumatic tire. Physical tire tests are

costly and difficult in operations. Thus, the parameters for the rigid ring tire model are

alternatively predicted by conducting virtual tire tests using a finite element analysis

(FEA) tire model.

A nonlinear three-dimensional FEA tire model representing a truck tire, 295/75R22.5, is

constructed by implementing three-layered membrane elements, hyperelastic solid

elements, and beam elements. Then, the FEA tire model is validated by comparing its in-

plane and out-of-plane responses with physical measurements. The virtual and physical

responses show good agreements. After successful validations of the FEA tire model,

virtual tire tests are conducted to predict the in-plane and out-of-plane parameters for the

rigid ring tire models for the first time using an FEA tire model.

The predicted parameters are implemented in the rigid ring tire model, and the model

undergoes water drainage ditches 90 and 45 to the tire running direction to predict

dynamic in-plane and out-of-plane tire responses at various tire loads. Vertical

displacement of the tire spindle, tire contact forces, and moments are plotted and

compared with those of the FEA tire model. The in-plane tire responses show good


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agreements between the results of the two models. On the other hand, the out-of-plane

tire responses are relatively not in good agreements due to the significantly different tire

contact area geometries of the two tire models on the 45ditch.

In the simulations of the FEA and rigid ring tire models, only constant vertical tire load is

applied to the tire models. Additional tire load due to the vertical acceleration of the

sprung mass during tire operations is not considered. Thus, a sprung mass and suspension

system is assembled with the tire models to include the effect of the vertical sprung mass

motion, which represents a quarter-vehicle model and a closer model to real vehicle

applications. Then, the models undergo a 90 ditch at various running speeds. The

vertical accelerations of the tire spindles are predicted during the ditch runs and

compared with measurements to check whether or not the rigid ring tire model in the

quarter-vehicle environment predicts acceptable responses.

Modern high computational capability enables to establish a reliable virtual tire and

quarter-vehicle model test environments. The developed quarter-vehicle model predicts

not only dynamic tire responses but also sprung mass responses to irregular road surface

inputs. Thus, a trustworthy quarter-vehicle model can replace the conventional field

durability tests, which saves product development cost and time. In addition, virtual testsunder similar conditions can be easily repeated. Higher quality products at lower cost are

undoubtedly promising.


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TABLE OF CONTENTS

LIST OF FIGURES ... x

LIST OF TABLES.. xiv

NOMENCLATURE.. xv

ACKNOWLEDGEMENT.. xvii

CHAPTER 1: INTRODUCTION. 1

1.1 History of the Wheel and Tire ... 3

1.2 Objectives .. 5

1.3 Methodology .. 6

1.4 Statement of the Work ... 7

CHAPTER 2: LITERATURE SURVEY. 10

2.1 Tire Axis System, Tire Forces, and Moments Definitions 10

2.1.1 Tire axis system ... 10

2.1.2 Tire forces and moments .. 11

2.1.2.1 Tire longitudinal force . 12

2.1.2.2 Tire lateral force 15

2.1.2.3 Tire vertical force .18

2.1.2.4 Tire overturning moment .. 20

2.1.2.5 Tire rolling resistance moment 21

2.1.2.6 Tire vertical moment 21

2.2 Tire Models 21

2.2.1 Early Tire Models ... 23

2.2.2 Finite Element Analysis Tire models ... 31

2.3 Contact and Sliding Interfaces .. 44

2.3.1 Computational Contact Solution Algorithms .. 44

2.3.1.1 Contact search algorithm . 46

2.3.1.2 Contact interface algorithm . 49

2.3.2 Lagrange Multiplier Method ... 50

2.3.3 Penalty Method ... 51

2.4 Experimental Tire Testing . 55


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2.5 Tire Rubber Material Modeling . 60

2.6 Summary 66

CHAPTER 3: NONLINEAR FINITE ELEMENT ANALYSIS TRUCK

TIRE MODELING.... 67

3.1 Tire Structure, Its Components, and Materials .. 69

3.1.1 Carcass 70

3.1.2 Belts 71

3.1.3 Tread and Treadbase ... 72

3.1.4 Beads .. 73

3.1.5 Aspect Ratio 73

3.2 Tire Structure and Material Modeling ... 75

3.2.1 FEA Modeling on Carcass and Belts Using Membrane

Elements ... 76

3.2.2 FEA Modeling on Tread, Treadbase, Tread Shoulders, Bead

Fillers Using Solid Elements 81

3.2.3 FEA Modeling of Beads 83

3.2.4 FEA Rim Model 85

3.3 FEA Tire-Rim Assembly Model 87

3.4 Various Virtual Tire Test Environments 88

3.4.1 Combined FEA Truck Tire and a Rigid Flat Road Model . 89

3.4.2 Combined FEA Truck Tire and a Rigid Cleat-Drum Model .. 90

3.4.3 Combined FEA Truck Tire and a Rigid Smooth Drum Model ... 91

3.4.4 Combined FEA Truck Tire and a Rigid Road with Water

Drainage Ditches Model . 92

3.5 Summary 92

CHAPTER 4: FEA TRUCK TIRE MODEL VALIDATION. 94

4.1 Static Validation Tests .. 94

4.1.1 Tire Vertical Load-Deflection Relationship ... 94

4.1.2 Footprint Area . 96

4.2 Dynamic Validation Tests . 98

4.2.1 First Mode of Free Vertical Vibration Test .... 99

4.2.2 Cornering Test 101

4.2.3 Yaw Oscillation Test .. 103

4.2.3.1 Amplitude Ratio and Phase Angle .... 104


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4.2.3.2 Virtual Yaw Oscillation Test Using FEA

Truck Tire Model .. 106

4.3 Summary 109

CHAPTER 5: DETERMINATION OF THE IN-PLANE AND OUT-OF-

PLANE PARAMETERS OF A RIGID RING TIRE MODEL

USING THE FEA TRUCK TIRE MODEL. 111

5.1 In-Plane Parameters Determination ... 112

5.1.1 Effective Rolling Radius . 114

5.1.2 In-Plane Translational Stiffness of the Sidewall and Residual

Vertical Stiffness at Contact Area ... 116

5.1.3 In-Plane Longitudinal and Vertical Damping Constants of

Sidewall and Residual Damping Constant at Contact Area 117

5.1.4 In-Plane Rotational Stiffnessand Damping Constant of the

Sidewall .. 118

5.1.4.1 In-Planed Rotational Stiffness of the Sidewall 119

5.1.4.2 In-Planed Rotational Damping Constant of the

Sidewall ... 119

5.1.5 Longitudinal Tread Stiffness and Longitudinal Slip Stiffness 121

5.1.5.1 Longitudinal Slip Stiffness .. 123

5.1.5.2 Longitudinal Tread Stiffness 123

5.1.6 Comparison of In-plane Tire Responses between the FEA and

Rigid Ring Tire Models .. 124

5.1.6.1 90Water Drainage Ditch Profile .... 125

5.1.6.2 Comparison of the Predicted Vertical Displacements . 125

5.1.6.3 Comparison of the Predicted Longitudinal Contact

Forces .. 127

5.1.6.4 Comparison of the Predicted Vertical Contact Forces . 129

5.2 Out-of-Plane Parameters Determination 131

5.2.1 Out-of-Plane Translational Stiffness and Damping Constant

of the Sidewall .... 132

5.2.2 Out-of-Plane Rotational Stiffness and Damping Constant

of the Sidewall .... 134

5.2.3 Lateral Free Vibration Tests ... 137

5.2.4 Cornering Stiffness . 139


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5.2.5. Self-Aligning Stiffness ... 139

5.2.6 Relaxation Length ... 140

5.2.7 Comparison of Out-of-Plane Tire Responses between the FEA

and Rigid Ring Tire Models ... 141

5.2.7.1 45Water Drainage Ditch Profile 142

5.2.7.2 Comparison of the Predicted Vertical Displacements . 143

5.2.7.3 Comparison of the Predicted Longitudinal Contact

Forces ... 144

5.2.7.4 Comparison of the Predicted Lateral Contact Forces .. 146

5.2.7.5 Comparison of the Predicted Vertical Contact Forces . 148

5.2.7.6 Comparison of the Predicted Overturning Moments ... 149

5.2.7.7 Comparison of the Predicted Vertical Moments .. 151

5.3 Summary 152

CHAPTER 6: QUARTER-VEHICLE MODELS USING RIGID RING

AND FEA TIRE MODELS.. 154

6.1 FEA and Rigid Ring Quarter-Vehicle Models .. 155

6.1.1 Sprung Mass ... 156

6.1.2 Front Axle Suspension Characteristics ... 156

6.2 Durability Tests Using a Tractor-Semitrailer

and Quarter-Vehicle Models . 157

6.2.1 90Water Drainage Ditch ... 158

6.2.2 Vertical Acceleration Measurements on the Ditch Using

Tractor-Semitrailer . 159

6.2.3 Vertical Acceleration Predictions on the Ditch Using

Quarter-Vehicle Models . 160

6.2.4 Model Validations ... 163

6.3 Summary 165

CHAPTER 7: CONCLUSION AND FUTURE WORK. 167

7.1 Conclusions 167

7.2 Major Contributions ... 169

7.3 Recommended Future Work .. 170

7.3.1 FEA Truck Tire Model in a Full-Vehicle Model 170

7.3.2 FEA Tire Model on a Soft Soil Terrain .. 171

7.3.3 FEA Vehicle/Tire Model for Hydroplaning Prediction ... 171


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REFERENCES 173

APPENDIX A: SIDEWALL DAMPING COEFFICIENT CALCULATION.. 183

APPENDIX B:.DIGITIZED DITCH PROFILES... 186


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LIST OF FIGURES

2-1 Wheel Axis, Forces, and Moments Definitions (Wong, 2001) .. 11

2-2 Major Components of Rubber Friction (Haney, 2004) .. 14

2-3 Cornering Force Developments at Various Slip Angles (Wong, 2001) . 16

2-4 Vehicle Under Cornering Maneuver and Lateral Force at the Contact Area . 17

2-5 Contact Area Shapes at Different Slip Angles (Haney, 2004) ... 18

2-6 Normal Pressure Distributions in the Contact Areas (Gillespie, 1992) . 19

2-7 Overturning Moment (Yap, 1991) .. 20

2-8 Tire Force Response Interactions (Allen et al., 1995) 23

2-9 Tire-Road Contact Mechanisms of the Early Tire Models

(Sui and Hirshey, 1999) 24

2-10 Equivalent Plane Tire-Road Contact Model (Sui and Hirshey, 1999) . 25

2-11 Five Degrees of Freedom of Lumped Mass-Spring Tire Model

(Takayama and Yamagishi, 1984) 26

2-12 In-Plane Rigid Ring Tire Model (Zegelaar and Pacejka, 1997) ... 28

2-13 Flexible Ring Tire Model (Kim and Savkoor, 1997) ... 29

2-14 Rigid Ring Tire Model (Bruni et al., 1997) . 30

2-15 Radial Tire Layout with a Belt Wedge near Tread Shoulder

(Kenny and Stechschulte, 1988) 34

2-16 Passenger Car Tire, P195/75R14, FEA Tire Model (Rhyne et al., 1994) 36

2-17 Inflated and Loaded Tire Model with Fixed Boundary Conditions in the Rim

Contact Areas (Zhang, 2001) . 39

2-18 Detail Tire Modeling of Isotropic and Anisotropic Layers (Zhang, 2001) .. 39

2-19 Normal Contact Pressure Distribution (Zhang, 2001) .. 40

2-20 Vibration Modes Test and Standing Waves Prediction (Chang, 2002) 41

2-21Two-Dimensional and Three-Dimensional Tire Models (Yan, 2003) .. 41

2-22 Tire Parts Subject to Relatively High Loadings (Yan, 2005) ... 43

2-23 Contact-Impact Simulation Methods (ESI, 2000) 45

2-24 Search Radius (ESI, 2000) 47

2-25 Node-to-Segment Correspondence Search (ESI, 2000) ... 48

2-26 Deformed Shapes Corresponding to the Pressures (Hallquist et al., 1985) ..53

2-27 Sequence of a Single-Surface Square Tube Buckling

(Benson and Hallquist, 1990) ... 54

2-28 Tire Tests on Internal and External Drums (Eldik Thieme, 1982) ... 56


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2-29 Test Wheel on Flat Rotating Disc (Kollmann, 1959) .. 56

2-30 Tire Test on Flat Belt (Eldik Thieme, 1982) 57

2-31 Randomly Oriented Chains (a) and Oriented Chains (b) (Askeland, 1994) 61

2-32 Principal Stresses and Extension Ratios (Yeoh, 1993) 62

2-33 Determination of the Mooney-Rivlin Coefficients (Yeoh, 1990) 65

3-1 Heavy-Duty Radial Truck Tire (Ford and Charles, 1988) . 69

3-2 Typical Tire Constructions (Wong, 2001) .. 70

3-3 Basic Tread Patterns of Truck Tires (Ford and Charles, 1988) . 72

3-4 Bead Bundle Configurations (Ford and Charles, 1988) 73

3-5 Definitions of a Tire Cross-Sectional Shape

(Tire and Rim Association Year Book, 1996) . 74

3-6 Radial Truck Tire (Retrieved from www.goodyear.com) . 76

3-7 Definition of Membrane Elements (ESI, 1998) 76

3-8 Three-Layered Membrane Element (ESI, 2000) ... 77

3-9 Carcass Wrap around Bead (Gough, 1981) 78

3-10 Locations of the Layered Membrane Elements and Their Part I.D.

Numbers in FEA Truck Tire Model . 78

3-11 Carcass and Belts Membrane Elements Used in FEA Truck Tire Model 80

3-12 Definitions of Solid Elements (ESI, 1998) .. 81

3-13 Locations of the Solid Elements and Their Part I.D.

Numbers in FEA Truck Tire Model 82

3-14 Solid Elements used in FEA Truck Tire Model .. 83

3-15 Definition of Beam Element 84

3-16 Beam Elements and Their I.D. Number in FEA Tire Model ... 84

3-17 Two Beads Used in FEA Truck Tire Model 85

3-18 15Drop Center Rim Contour (Tire and Rim Association Year Book, 1996). 86

3-19 FEA Truck Tire Rim Model . 87

3-20 FEA Tire-Rim Assembly Model .. 88

3-21 FEA Truck Tire on a Rigid Flat Road Model .. 89

3-22 FEA Truck Tire at 4 Steer Angle on a Rigid Flat Road Model ...... 89

3-23 FEA Truck Tire on a Rigid Cleat-Drum Model ... 90

3-24 FEA Truck Tire on a Rigid Smooth Drum Model ... 91

3-25 FEA Truck Tire on a Rigid Flat Road with the Three Water Drainage Ditches92

4-1 Virtual Tire Load-Deflection Test . 95

4-2 Vertical Tire Load versus Deflection . 95

4-3 Footprint Areas and Shapes of Various Tires (Tekscan, 2006) . 96


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4-4 Footprint Area Prediction at Various Tire Loads ... 97

4-5 Footprint Area versus Vertical Static Load 98

4-6 In-Plane Tire Excitation During Cleat Impact 99

4-7 FFT Result of Vertical Reaction Force at Tire Spindle .. 100

4-8 First Mode of Vertical Free Vibration (Gillespie, 1992) 100

4-9 Cornering Simulation at Slip Angle of 6.. 101

4-10 Cornering Forces Validation 102

4-11 Aligning Moments Validation .. 102

4-12 Yaw Oscillation Test 103

4-13 Yaw Oscillation Input and Lateral Force Output at the Frequency of 1 Hz . 107

4-14 Tire Lateral Force Magnitude as a Function of Path Frequency .. 108

4-15 Tire Phase Angle as a Function of Path Frequency .. 108

5-1 In-Plane Rigid Ring Tire Model (Zegelaar and Pacejka, 1997) . 113

5-2 Tire Deflection during Rolling ... 114

5-3 Effective Rolling Radius, Re .. 115

5-4 Rotational Excitation on FEA Truck Tire .. 118

5-5 Angular Displacement of the Tread and Damping Response of the Sidewall ... 119

5-6 Longitudinal Force versus Slip Ratio . 122

5-7 90 Water Drainage Ditch Profile ... 125

5-8 Tire Spindle Vertical Displacements at Tire Load of 13.3 kN ... 126

5-9 Tire Spindle Vertical Displacements at Tire Load of 26.7 kN ... 126

5-10 Tire Spindle Vertical Displacements at Tire Load of 40.0 kN . 126

5-11 Longitudinal Contact Force at Tire Load of 13.3 kN ... 128



5-14 Vertical Contact Force at Tire Load of 13.3 kN ... 129



5-17 Out-of-Plane Parameters for the Rigid Ring Tire Model (Volvo 3P, 2004) 131

5-18 Out-of-Plane Translational Excitation on Sidewall . 132

5-19 Lateral Displacement and Damping Response of the Sidewall 133

5-20 Out-of-Plane Rotational Excitation on Sidewall .. 135

5-21 Out-of-Plane Rotational Displacement and Damping Response

of the Sidewall .. 135

5-22 Lateral Excitation at Tire Spindle . 137

5-23 Lateral Free Vibration Test Result at Vertical Tire Load 26.7 kN ... 137


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5-24 Cornering Stiffness Prediction at Vertical Load of 26.7 kN 139

5-25 Self-Aligning Stiffness Prediction at Vertical Load of 26.7 kN .. 140

5-26 45 Water Drainage Ditch Profile . ... 142




5-30 Longitudinal Contact Forces at Tire Load of 13.3 kN . 145

5-31 Longitudinal Contact Forces at Tire Load of 26.7 kN ..145

5-32 Longitudinal Contact Forces at Tire Load of 40.0 kN . 146

5-33 Lateral Contact Forces at Tire Load of 13.3 kN ... 147



5-36 Vertical Contact Forces at Tire Load of 13.3 kN 148

5-37 Vertical Contact Forces at Tire Load of 26.7 kN . 149

5-38 Vertical Contact Forces at Tire Load of 40.0 kN 149

5-39 Tire Spindle Overturning Moment at Tire Load of 13.3 kN 150



5-42 Tire Spindle Vertical Moments at Tire Load of 13.3 kN 151

5-43 Tire Spindle Vertical Moments at Tire Load of 26.7 kN . 151

5-44 Tire Spindle Vertical Moments at Tire Load of 40.0 kN . 152

6-1 Quarter-Vehicle Models . 155

6-2 Spring Characteristics of the Suspension 156

6-3 Shock Absorber Characteristics of the Suspension 157

6-4 Measurement of the 90 Ditch Profile 158

6-5 90 Ditch Profile .. 158

6-6 Test Tractor-Semitrailer Running over the Ditch .. 159

6-7 Accelerometer Location at Front Left Wheel 159

6-8 Tire Spindle Vertical Accelerations at 8.2 km/h (Rated Tire Load: 26.7 kN) 160

6-9 Tire Spindle Vertical Accelerations at 11.1 km/h (Rated Tire Load: 26.7 kN) . 161

6-10 Tire Spindle Vertical Accelerations at 21.3 km/h (Rated Tire Load: 26.7 kN) 161

7-1 FEA Tire Model on a Soft Soil Terrain .. 171

7-2 FEA Vehicle/Tire Model for Hydroplaning Prediction .. 172


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LIST OF TABLES

3-1 Material Properties of the Layered Membrane Elements ... 79

3-2 Material Properties of the Solid Elements .. 83

3-3 Material Properties of the Beam Elements . 84

3-4 15Drop Center Rim Contour Dimensions for 22.5 8.25 Truck Tire Rim . 86

3-5 Material Properties of the FEA Rim .. 87

5-1 In-plane Translational and Residual Vertical Stiffness of Sidewall

at Tire Load of 26.7 kN .117

5-2 In-plane Longitudinal and Vertical Damping Constants of the Sidewall and

Residual Damping Constant at Contact Area at Tire Load of 26.7 kN .. 118

5-3 In-plane Rotational Stiffness and Damping Constants of Sidewall

at Tire Load: 26.7 kN 121

5-4 Predicted Rigid Ring Tire Model Parameters at Given Tire Load . 124

5-5 Parameters Used for the Out-of-Plane Translational Damping Constant .. 134

5-6 Parameters Used for the Out-of-plane Rotational Damping Constant ... 136

5-7 Parameters Used for the Out-of-plane Slip Damping Constant . 138

5-8 Out-of-Plane Characteristic Parameters of the Rigid Ring Tire Model . 141

6-1 Statistical Validation of Tire Spindle Vertical Acceleration at 8.2 km/h ... 165

6-2 Statistical Validation of Tire Spindle Vertical Acceleration at 11.1 km/h . 165

6-3 Statistical Validation of Tire Spindle Vertical Acceleration at 21.3 km/h . 165


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NOMENCLATURE

Symbol Description Unit

a Half contact length between tire and road surface mC10, C01 Mooney-Rivlin coefficient N/m

2

cbx, cbz In-plane translational damping of sidewall N s/m

cby Out-of-plane translational damping constant N s/m

cb Out-of-plane rotational damping constant N m s/rad

cb In-plane rotational damping of sidewall N m s/rad

cc Critical damping constant N s/m

cl Out-of-plane slip damping constant N s/m

cvr Residual damping constant N s/m

d Tire Deflection due to loading m

f Stretching force on a single chain molecule N

fr Rolling resistance coefficient -Fx Longitudinal or tractive force N

Fy Lateral force N

fy Yaw oscillation frequency Hz

Fy_R Resultant lateral force N

Fz Vertical or normal force N

Fz_R Resultant vertical force N

Iax, Iaz Mass moment of inertia of wheel rim about X- and Z-axis kg m2

Iay Mass moment of inertia of wheel rim about Y-axis kg m2

Ibx, Ibz Mass moment of inertia of tire belt about X- and Z-axis kg m2

Iby Mass moment of inertia of tire belt about Y-axis kg m2

Ii Strain invariant (i=1, 2, and 3) -kbx, kbz In-plane translational stiffness of sidewall N/m

kby Out-of-plane translational stiffness N/m

kb Out-of-plane rotational stiffness N m/rad

kb In-plane rotational stiffness of sidewall N m/rad

kcx Longitudinal tread stiffness N/m

kf Cornering stiffness N/rad

kk Longitudinal slip stiffness N/slip unit

kL Lateral tire stiffness N/m

kl Lateral slip stiffness N/m

kM Self-aligning stiffness kN m/rad

ktot Tire total vertical stiffness N/mkvr Residual vertical stiffness N/m

ma Wheel rim mass kg

mb Tire belt mass kg

Mx Overturning moment N m

My Rolling resistance moment N m

Mz Vertical or aligning moment N m


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Symbol Description Unit

R Radius of the inflated tire before loading m

Re Effective rolling radius m

Rdrum Drum radius m

s Longitudinal offset between the wheel center and Fz_R m

T Absolute temperature Kti Principal stress (i=1, 2, and 3) N/m

2

u Current wheel orientation -

v, vtire Tire speed m/s

vdrum Drum speed -

vtr Tread speed -

W Strain energy density function -

Slip angle rad

Phase angle rad

Amplitude ratio of the output to yaw oscillation input -

Lagrange multiplier N

i Extension ratio (i=1, 2, and 3) -

System time constant s

d Damped period of vibration s

Wheel angular speed rad/s

d Damped natural frequency rad/m

drum Drum angular speed rad/s

n Undamped natural frequency rad/m

path Path frequency rad/m

y Yaw oscillation frequency rad/s

Damping ratio -


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ACKNOWLEDGMENTS

For many years, obtaining a doctoral degree in engineering has been my highest

professional goal. Taking the steps to achieve this goal has proved challenging yet

rewarding. During the process, many have influenced and contributed to my development

both as a student and as a person. Special thanks are owed to the members of my graduate

committee: Dr. James P. Runt, Dr. Ashok D. Belegundu, and Dr. Charles E. Bakis. Each

of you has provided invaluable research and teaching guidance in pursuit of my

professional ambition.

Dr. Moustafa El-Gindy, chair of my committee, has provided unwavering support andleadership throughout my academic career. His professional accomplishments are

inspiring and his strong encouragement has helped me remain focused and motivated

throughout my tenure at Penn State. I am deeply indebted to Dr. El-Gindy for his time

and thoughtful consideration.

I also thank the Goodyear Tire and Rubber Company for providing the measurements of

the tire deflection and footprint area versus vertical tire loads. I would like to express my

deep appreciation to Mr. Trivedi Mukesh and Mr. Fredrik ijer of Volvo 3P, the sponsor

of the research project, for their continuous technical help during the course of this study.

On a personal note, I would like to give thanks to my parents, Dr. Jungmin Chae and

Taesun Son, for their endless guidance and care. They have taught me hard work,

dedication, and commitment. The personal opportunities that they have provided would

take more than a lifetime to repay. Also, I would like to thank Dr. Benedict Y. Oh and

MD. Susan Oh, my parents of this town, State College, for their cordial encouragementand care. I also thank my brother, Seokhyun, and sisters, Sungwon and Sunghee, for their

encouragement during this endeavor.


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Lastly, I would like to express my sincere appreciation to my wife, Eunju Kim. She has

devoted herself to support and encourage me even though she is busy in pursuing her

doctoral degree. Thus, I dedicate this thesis to my wife, Eunju Kim. Without the patience,

love, and understanding of my wife, this accomplishment would not have been possible.


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CHAPTER 1

INTRODUCTION

Tires are among the most essential components of ground vehicles. They perform many

important functions during vehicle operation. For example, they support vehicle weight

enough according to its own rated load capacity. They also transmit sufficient driving,braking, and cornering efforts between the rim and road surfaces. They have the ability to

resist the longitudinal, lateral, and vertical reaction forces from the road surface without

severe deformation or failure. Further, they also alleviate shocks from road surface

irregularities to a certain degree due to their damping and energy dissipation nature.

Eventually, tires provide a safe and comfortable environment for passengers and luggage.

If tires cannot perform all these tasks properly, the driver may easily lose control of the

vehicle and face serious safety problems.

In order to satisfy these performance requirements mentioned above, tires need to be

robust enough to withstand the applied vertical wheel load, frictional shear forces, and

wear generated on the tire-road contact area. At the same time, tires need to be soft and

flexible enough to absorb shocks due to road surface irregularities. Because there does

not exist currently a single engineering material to serve for all these performance

requirements of the tires, steel-cords embedded rubber composite materials are used. As

can be observed, the tire structure is complex to fulfill the performance requirements and

challenging to design and manufacture.

For the last 120 years since the first pneumatic tire was invented, many technologies have

been continuously developed to satisfy the performance requirements and enhance the


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quality of tires. Higher quality of passenger car tires has been developed to provide a

softer and more comfortable ride for passengers. Simultaneously, stronger commercial

vehicle1tires have been also developed to withstand and carry higher loads of freight.

Whenever a new type of tire is designed and manufactured, tire testing is required to

characterize the performances of the new tire. Many tire testing set-ups have been

developed to measure static and dynamic tire responses in a laboratory or a test field.

However, the experimental tire testing is usually costly and difficult to build. In addition,

experiment equipment, its set-up, data acquisition, and analysis need highly experienced

skills and long testing time. Sometimes, the experimental tire testing is governed by

weather and temperature of the test field environment. In addition, some extreme cases

such as high tire loading and/or high speed of tire rotation cannot be conducted by usingconventional testing equipment. It can also take large amount of time and effort to repeat

same or similar tire tests. In order to overcome these limitations of the experimental tire

testing, many researchers have tried to build alternative tire testing environments during

the last few decades.

Fortunately, modern computer technology enables to open a new era of tire testing.

Through tire model simulations, most of the laboratory tire tests can be duplicated. Even

limited tire tests that cannot be performed in laboratory, such as high speed and/or

loading operations, are possible with the tire model simulations. Among tire models, a

rigid ring and a finite element analysis (FEA) tire models are widely used. The rigid ring

tire model is simple but needs several in-plane and out-of-plane parameters to be

determined through comprehensive tire tests. The rigid ring tire model is simulated not

only alone but also together in multi-body vehicle model environment. On the other hand,

the FEA tire model is complicated to build but versatile in its applications.

In this study, the parameters of the rigid ring tire model are predicted by using an FEA

truck tire model, then, the rigid ring tire model is tested on a durability test event alone

and in a quarter-vehicle environment. The predicted dynamic tire responses of the rigid

1Buses, trucks, trailers, and moving vans are categorized as the commercial vehicles.


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ring tire model are compared with those of the FEA truck tire model and measured values

using a tractor-semitrailer.

1.1 History of the Wheel and Tire

In the Paleolithic era, 15,000 to 750,000 years ago, people used round logs to move

heavy objects more easily (The Evolution of the Wheel1, 2006). They placed logs and a

sledge under a heavy object and dragged the sledge over one log to the next. As the

sledges started to wear grooves into the log between the two ends, the log gradually

changed shape similar to a big bone. Then, the wood between two ends was cut to create

the axle. The two ends finally became solid wheels even though they were primitive. The

first use of the solid wheel for transportation was assumed to be used on Sumerian 2

chariots in 3,500 BC (Dillion and Rockefeller, 1999). The two-wheeled chariot was

found in the birthplace of Sumeria and is believed to be the first form of wheeled

transportation. This chariot increased the speed of travel over land and eventually led to a

four-wheeled cart, which took the burden of carrying supplies and equipment off the

shoulders of men.

For over a few thousands years, wooden wheels had been manufactured with different

spokes, leather-tops, and iron strip-tops, in chronological order. In 1839, Charles

Goodyear discovered the vulcanization process (Goodyear, 1853), which is the process of

heating raw rubber with sulfur to transform sticky raw rubber to a firm but pliable

material. The process makes rubber a perfect material for tires and other engineering

applications. Soon after the discovery of the vulcanization process, bicycle tires could be

manufactured using the vulcanized solid rubber. These solid rubber tires were strong

enough to resist cuts and abrasions. They could even absorb some amount of shocks

1http://www.ohtm.org/wheel.html2Sumeria (or Sumer) is one of the first civilized societies in southern Mesopotamia areafrom around 3,800 BC to 2,000 BC.


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from roadway irregularities. However, despite these safety features, these tires were still

very heavy and did not provide a smooth ride to driver.

Later, in 1845, Robert William Thomson, a Scottish engineer, conceived an idea of air-

inflated or pneumatic bicycle tires and expected that the pneumatic tires could overcome

the limitation of the solid rubber tires (Bellis, 2006). However, his idea of a pneumatic

tire was only patented as a concept and was never carried out to manufacture a real

pneumatic tire. Meanwhile, John Boyd Dunlop, a Scottish veterinarian, watched his son

encountered difficulty riding a tricycle over cobbled ground 1 . He realized that the

unforgiving ride was due to the solid rubber tires of the tricycle. He started to find a way

to improve the ride performance of the tires. Shortly, he considered that an air-inflated

tire would be light and absorb more shocks that provided a smoother ride. Eventually, hemanufactured the first pneumatic bicycle tire in 1888 and patented it, which replaced

most of the solid bicycle and tricycle tires within ten years.

One day in 1889, a bicyclist brought a punctured bicycle tire to the Michelin brothers,

Andr and douard Michelin, to fix (Automotive News Europe, 2001). They found a

major drawback of Dunlops tire in that the tire was firmly glued to the rim, which made

punctured tire repair very difficult. The Michelin brothers decided to help the bicyclist

and tried to find an easier method to repair punctured tires. Eventually, they

manufactured a detachable pneumatic tire that could save time and effort to repair a

punctured tire. A few years later, Michelin brothers made an effort to convince carmakers

of the utility of inflatable tires. At that time, cars had used the same kind of wheel as

those used on horse-drawn vehicles which was a wooden wheel with a metal rim or a

solid tire. The brothers effort was accepted and, within a few years, the Michelin firm

achieved astonishing growth by serving the early stage of the automotive industry.

1http://www.tartans.com/articles/famscots/jbdunlop.html


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1.2 Objectives

For an efficient full vehicle model simulation, a multi-body system (MBS) simulation is

frequently adopted. In the MBS simulations, a rigid ring tire model is mostly used due toits efficiency. For the rigid ring tire model, several in-plane and out-of-plane parameters

need to be determined through tire tests to represent a real pneumatic tire. The physical

tire tests are costly and difficult in operations. Thus, the parameters for the rigid ring tire

model are alternatively predicted by conducting virtual tire tests using an FEA truck tire

model. The rigid ring tire model implemented with the determined parameters will

generate tire responses in a multi-body truck model close to those of an actual tire, thus, it

will contribute to obtain the efficient results of vehicle model simulations.

The first objective of this study is to build a reliable truck tire model and a virtual tire

testing environment to replace the conventional expensive and limited physical tire tests.

An FEA truck tire model is created based on a real truck tire, 295/75R22.5, dimensions

and material data. Then, this tire model experiences validation tasks to check whether it

follows the similar behaviors of the available measured data.

As the second objective of the study, in-plane and out-of-plane characteristic parametersfor a rigid ring tire model to represent the truck tire are predicted by using the developed

FEA tire model and virtual testing environments at various tire operation conditions.

After the predicted parameters are implemented in the rigid ring tire model, the model is

tested by running on water drainage ditches. The predicted in-plane and out-of-plane tire

responses are compared with those of the FEA truck tire model.

As the third objective of the study, a quarter-vehicle model using the rigid ring tire model

is created to test the rigid ring tire model under the influence of the sprung mass vertical

acceleration during the ditch runs. The predicted vertical accelerations at the tire spindle

are compared with measurements and results from FEA quarter-vehicle model for the

model validation purposes and will serve for handling, ride quality, stability, and safety

analyses of the vehicle.


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Tire mechanics is quite complex to analyze because of highly nonlinear dynamic

characteristics intrinsic to its nature. The establishment of a precise and efficient model is

of critical importance to the analysis, simulation, and design of safer and more advanced

tires. Once the reliable virtual tire testing environment has been established, tire model

simulations will be easily repeated at various operation conditions, even severe operation

conditions that cannot be accomplished in laboratory testing facilities.

1.3 Methodology

For the last 50 years, many tire models have been established to simulate transient and

steady-state behaviors on various road terrains. For example, flexible ring, rigid ring,

point contact, and FEA tire models have been performed to predict the tire responses.

Among those tire models, the FEA tire model has performed the most reliable and

versatile tasks under various operation conditions. Thus, the FEA is adopted in this study

to build the truck tire model and testing environments. The FEA computer simulation

software, PAM-SHOCK, is used to accomplish the tire modeling and virtual testing.

Finally, the FEA tire model is completed with a validation task. The validation task of themodel involves mainly checking whether the FEA model yields a reasonable behavior as

a real tire behaves in a real application. For the validation of the tire model, the predicted

results from the vertical load-deflection, footprint area, first free vertical vibration mode,

cornering characteristics, and yaw oscillation tests will be compared with physical tire

test measurement data. The measurement data are provided by the Goodyear Tire and

Rubber Company and the University of Michigan Transportation Research Institute

(UMTRI).

Subsequently, the developed tire model will be tested to predict characteristic tire

parameters at various operation conditions, such as various vertical tire loads, speeds, and

steering inputs. The tire model will be inflated at rated pressure and loaded at a given tire

load on a road surface. After the tire model is stabilized, the tire model will be excited


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according to the test specifications. Durability tests will also be performed on water

drainage ditches using a tire model only and quarter vehicle model to predict vertical

responses of the tire spindle. For this durability test, the FEA mesh model of 86 mm-deep

and 736 mm-long water drainage ditch will be created. After the virtual durability test,

the predicted vertical responses will be compared with real measurement data provided

by one of the leading truck manufacturers.

1.4 Statement of the Work

Chapter 2 presents a comprehensive literature survey on tire models to help the

understanding of tire mechanics and modeling techniques. In addition, the direction of

this study can be guided by conducting the literature survey. Tire models are categorized

in three different major methods: analytical tire model, rigid ring tire model, and FEA tire

model. The modeling techniques of those models and their applications are summarized.

The advantages and limitations of each modeling method are also described. The

literature review focuses more extensively on the rigid ring tire models and FEA tire

models, which cover most of this research. At the end of Chapter 2, literature on the

calculation of the Mooney-Rivlin coefficients follows. The method to determine thecoefficients from rubber material tensile and compression tests is described in detail.

In Chapter 3, a three-dimensional nonlinear FEA truck tire model, 295/75R22.5,

including a detailed rim is constructed using the FEA simulation software, PAM-SHOCK.

The methods and techniques to model tire structure, its components, and materials are

described in detail. Beam elements, layered membrane elements, and solid elements are

used to build the truck tire model. In addition, various virtual tire tests environments to be

used to test the developed FEA tire model are presented at the end of the chapter. The

FEA truck tire model will be used to predict in-plane and out-of-plane tire parameters of

a rigid ring tire model after the validation tasks.


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In Chapter 4, validation tasks of the developed FEA truck tire model are conducted

through virtual static and dynamic tire tests. The static tests include vertical tire load-

deflection and footprint area prediction tests. The dynamic tests consist of the first free

vertical vibration mode, cornering, and yaw oscillation tests. The first free vertical

vibration mode test is conducted on a rotating cleat-drum, and it is obtained by applying

FFT algorithm to the predicted vertical force time history at the tire spindle. For the

validation of the model in terms of steady-state out-of-plane motions, cornering test

simulations are conducted at various slip angles and tire loads to predict cornering forces

and aligning moments applied on the tire model during the tests. Yaw oscillation test is

also conducted to examine the dynamic out-of-phase tire responses to sinusoidal steering

angle input. As a result, the amplitude ratio and phase angle are plotted as a function of

the path frequency. These predicted static and dynamic tire responses agree with physicalmeasurements.

Chapter 5 presents the determination of the in-plane and out-of-plane parameters for a

rigid ring tire model through virtual FEA tire tests. The in-plane parameters are the

translational stiffness and damping constant of the sidewall along the longitudinal and

vertical axes, the rotational stiffness and damping constant of the sidewall about the

lateral axis, and the vertical stiffness and damping constant of the tire belt and tread. The

out-of-plane parameters are the translational stiffness and damping constant along the

lateral axis and the rotational stiffness and damping constant of the sidewall about the

longitudinal axis. These parameters are predicted at a tire inflation pressure of 0.759 MPa

under three tire loading cases, 13.3 kN (3,000 lb), 26.7 kN (6,000 lb), and 40.0 kN (9,000

lb). The predicted tire parameters are implemented into a rigid ring tire model, and the

tire model is tested on 90and 45water drainage ditches to predict in-plane and out-of-

plane tire responses at a tire speed of 19.3 km/h (12 mph) and the various tire loads.

Vertical tire displacements, applied forces, and moments applied on the tire model are

predicted. The predicted responses of the rigid ring tire model are compared with those

from the FEA tire model simulations.


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Chapter 6 presents a quarter-vehicle model to test the rigid ring tire model in an

environment that includes the effect of the vertical motion of the sprung mass on the

dynamic responses of the tire models. Thus, the tire model in the quarter-vehicle model

generates closer dynamic tire responses to real tire behaviors. The rigid ring quarter

vehicle model consists of the rigid ring tire model, suspension, and sprung mass. For the

suspension, the spring and shock-absorber characteristics of a tractor front axle are

implemented to connect the sprung mass and the tire model. The sprung mass is

constrained to move only vertically. The quarter vehicle model runs on the 90ditch to

predict the vertical acceleration at the tire spindle at various tire running speeds. Then,

the predicted vertical accelerations are compared with those from the FEA quarter-

vehicle model and measured values on the front axle of the tractor-semitrailer. The rigid

ring quarter-vehicle model simulations and measurements are conducted by one of the

leading truck manufacturers.

In Chapter 7, the conclusions obtained throughout this study are summarized. The major

contributions, advantages, and limitations of the FEA tire model are also discussed. For

future work, the dynamic performances of the FEA truck tire model in a full vehicle

model, FEA tire model responses on a soft terrain, FEA vehicle/tire model for hydro-

planning prediction are recommended.


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CHAPTER 2

LITERATURE SURVEY

In this chapter, literature previously published on various tire modelings and analysis has

been reviewed to help historical understanding and to provide tire modeling methods

including their capabilities and limitations. In addition, the reviews on the various

solution algorithms of the computational contact problems, tire testing, and tire rubber

material modeling are followed. Before various tire modeling methods and the other

topics are introduced, reference axis system, definitions of forces and moments applied

on the tire are described and are later frequently mentioned throughout this study.

2.1 Tire Axis System, Tire Forces, and Moments Definitions

Section 2.1 describes a conventional tire axis system, forces, and moments definitions

that are widely accepted. Also, according to the axis system, the forces and moments that

can be applied on the tire during operation are defined. Three forces and three moments

are described.

2.1.1 Tire axis system

In order to describe the characteristics of a tire and applied forces and moments on the

tire, it is necessary, first, to define an axis system that serves as a reference for the

definition of various parameters. One of the commonly used axis systems is

recommended by the Society of Automotive Engineers (SAE) as shown in Figure 2-1.

The origin of the axis system is the center of tire contact. The X-axis is the interaction of


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the wheel plane and the ground plane with a positive direction forward. The Z-axis is

perpendicular to the ground plane with a positive direction downward. The Y-axis is in

the ground plane and its direction is chosen to make the axis system orthogonal, which is

on the right of the wheel plane.

Figure 2-1 Wheel Axis, Forces, and Moments Definitions (Wong, 2001)

2.1.2 Tire forces and moments

Since the tire is the only media in a vehicle that has contact with road surface, the tire

needs to support sprung mass, resist external disturbances, and transmit driving and

braking torques from the vehicle to the road surface. Therefore, forces and moments are

continuously applied to the tire along all three axes during vehicle operations as shown in

Figure 2-1. In this section, three forces and three moments are described in detail with

their origins and important functions. The forces are longitudinal or tractive force (Fx),

lateral force (Fy), and vertical or normal force (Fz). The moments are overturning moment

(Mx), rolling resistance moment (My), and vertical moment or aligning moment (Mz).

These forces and moments are mainly due to the tire-road contact and its elastic

deformation. When the tire doesnt have contact with road surface, no forces and

moments will be applied to the tire.


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2.1.2.1 Tire longitudinal force

When a tire is rolling or sliding along the longitudinal direction on a road, longitudinal

force is applied to the tire at the contact area. The longitudinal force can be categorized

into A) the rolling resistant force, B) longitudinal frictional force, and C) longitudinal

reaction force according to their causes.

A) Rolling resistant force

During free rolling1along a straight direction, the rolling resistant force is applied to the

tire at the contact area against the tire rolling direction. The rolling resistant force or

rolling resistance of tires is primarily caused by the hysteresis in tire materials due to the

carcass deflection of rolling tires. In addition to the hysteresis, many other factors can

affect the rolling resistance of a pneumatic tire. The factors are known to be tire

construction, materials, and its various operating conditions such as road surface

condition, inflation pressure, speed, and temperature. At rated load and inflation pressure

on the same size tires, bias-ply tires show higher rolling resistance than radial-ply tires

due to greater hyteresis losses in bias-ply tires. Thicker treads, thicker sidewalls, and

increased number of carcass plies also tend to increase the rolling resistance due to

greater hyteresis losses. Tires made of synthetic runner compounds generally have higher

rolling resistance than those made of natural rubber.

Surface conditions also affect the rolling resistance. On hard and smooth surfaces, the

rolling resistance is significantly lower than that on a soft and rough road. Each road

surface condition shows the corresponding coefficient of rolling resistance which is

defined as the rolling resistant force divided by vertical tire load. For example, for

passenger car tires, the coefficient of rolling resistance is 0.013 on a concrete or asphalt

road and 0.05 on an unpaved road. For general truck tires, the coefficient of rolling

resistance is 0.006-0.01 on a concrete or asphalt road, which is lower than that for

passenger car tires due to larger tire diameter and higher inflation pressure. On wet

1 Free rolling condition is defined as the rolling condition without traction or braking

torques applied on a tire.


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surfaces, a higher rolling resistance is usually observed than on dry surfaces. In addition,

a higher rolling resistance is also observed at higher speed, lower inflation pressure, and

lower internal tire temperature.

B) Longitudinal frictional force

In acceleration and braking operations, there is speed difference between the rolling

speed of a tire and its traveling speed, which results in a certain degree of slip between

tire tread and road surface. Without acceleration and braking efforts, no slip will occur.

With a certain amount of slip, a frictional force is developed in the tire-road contact area

that enables the vehicle to be accelerated and decelerated. In the case of a braking

operation, the degree of the slip can be expressed in slip ratio according to the equation

(2-1).

Slip ratio (%) = 100)1(

v

eR (2-1)

Where,Re: tire effective rolling radius,

: wheel angular velocity,

v: tire speed.

When the braking effort is so high that the wheel is locked and slides on the road surface,

the slip ratio is defined as 100%. Normally, the slip ratio at which the frictional force

reaches maximum falls between 10% and 30%. Therefore, the anti-skid brake system

(ABS) maintains the slip ratios always between 10% and 30% to generate maximum

braking efforts at full braking maneuvers.

Tire tread rubber generates friction in three major ways: adhesion, deformation, and

tearing/wear. Figure 2-2 shows these three components that contribute to the total friction

force experienced by tread rubber on the road surface at a slip speed ofv.


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Vertical Load

=

Aggregate

++

Figure 2-2 Major Components of Rubber Friction (Haney, 2004)

Surface adhesion arises from the momentary intermolecular bonds between the tread

rubber and the aggregate in the road surface. The adhesion depends on the true contact

area that is determined by the road surface profile, involved material properties, and

contact pressure. Normally, the adhesion component is the major contributor in tire

traction on dry and smooth roads. However, when the road is contaminated with dust or

water, the tire loses part of the contact and formation of adhesive forces. Then, the

adhesion friction is reduced substantially, which results in the loss of friction.

Tread rubber in contact with a smooth surface generates friction force mainly byadhesion. However, when rubber is in contact with a rough road surface, another

mechanism, deformation, plays an important role in friction. As the tread rubber slides on

a rough road surface, the local deformations of the rubber are observed on road surface

irregularities. Friction forces due to those local deformations provide most of the friction

force between the tire and wet road surface.


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In addition to the adhesive friction and deformation friction, the rubber produces friction

forces by means of tear and wear. As the applied load and sliding speeds increase, local

stress can exceed the tensile strength of the rubber especially near the area of a sharp

irregularity. High local stress can deform the internal structure of the rubber beyond the

point of elastic recovery. When the polymer chains are stressed to failure, tearing may

occur. Tearing absorbs energy and results in additional friction forces at the contact

surface.

The friction force is different from the rolling resistant force. The rolling resistant force is

applied on the freely rotating tire whereas the friction force is applied under acceleration

or deceleration operations where slip exists. When the tire is under acceleration due to

driving torque, the tractive force is applied in the direction of the motion. Conversely,

when the tire is under deceleration due to braking, the braking force is applied to the tire

against the tire moving direction. However, the rolling resistance is always applied

against the tire moving direction.

C) Longitudinal reaction force

When the tire runs over severe road surface irregularities or obstacles such as steps,

potholes, water drainage ditches, or speed bumps, a reaction force is applied to the tire

longitudinally as well as laterally and vertically. This longitudinal reaction force is

usually applied as a shock, which can cause damage to the tire and rim. All of these

longitudinal forces are usually acting opposite to the tire moving direction at the tire-road

contact area. The reaction force depends significantly on the suspension characteristics of

the vehicle and tire operational conditions such as vertical load, inflation pressure, and

speed.

2.1.2.2 Tire lateral force

When a vehicle undertakes a cornering operation, or subjected to cross-wind, lateral force

is developed at the tire-road contact area. The lateral forces during a cornering maneuver

and under irregular lateral wind are dynamic forces due to the lateral acceleration of the


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vehicle. The lateral force in reaction to a cornering maneuver is called a cornering force.

The cornering force is highly dependent of tire vertical load. As the vertical load on the

tire increases under the same cornering operational condition, the cornering force also

increases. Meanwhile, during cornering maneuvers, a higher vertical load is exerted on

the right tires due to lateral load transfer. Therefore, higher cornering forces are also

applied to the same right tires.

In addition, the cornering force also depends on the slip angle of the tire. As the slip

angle increases under the same vertical load on the tire, the cornering force also

increases. However, when the cornering force reaches a certain level, it does not

significantly increase further. Instead, it tends to converge to an asymptote, which is a

road surface adhesion limit as seen in Figure 2-3.

Figure 2-3 Cornering Force Developments at Various Slip Angles (Wong, 2001)

The formation of the slip angle is attributed to the elastic nature of the tire. The tire tread

grips the road surface due to friction. However, the tire also resists movement with an

opposing force, yields to external force, and recovers when the external force is removed.

This elastic characteristic allows the tire to have an orientation (u) different from the

direction in which the vehicle is traveling (v) as shown in Figure 2-4 (a). The angle, ,


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between the two orientations is defined as slip angle and plays an important role in

cornering operations.

A: Front left wheel

CG: Vehicle center of gravity

O: Center of the front left wheelPQ: Cornering curve path

u: Current wheel orientationv: Vehicle heading direction from the CG

v: Vehicle heading direction from thewheel A

: Slip angle

(a) Vehicle under cornering maneuver with slip angle

Fy_R

PT

O

A

(b) Lateral force distribution at tire-road contact area during cornering maneuver

Figure 2-4 Vehicle Under Cornering Maneuver and Lateral Force at the Contact Area

The developed cornering force at the contact area is illustrated in detail in Figure 2-4 (b).

When the vehicle is turning left, a cornering force is applied on the tire to the left at the

contact area. However, the cornering force is not distributed symmetrically on the contact

area around the tire center. Instead, the peak cornering force moves behind the center of


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the tire due to the longitudinal force against the tire. Therefore, the resultant cornering

force, Fy_R, can be drawn behind the tire center with offset, PT, as shown in Figure 2-4

(b). The offset is called the pneumatic trail (PT). This pneumatic trail and the resultant

cornering force generate an aligning moment about the vertical axis.

The schematic contact area shapes are illustrated in Figure 2-5 at different slip angles.

Originally, the tire is rolling in the direction of the top of the page and is currently turning

left. As a result, effective stationary contact (adhesive area) occurs always at the leading

edge of the contact area as the tire rolls. On the other hand, slip is confined to the rear of

the contact area because the resultant cornering force is applied behind the tire center. In

the region of the slip, the tangential surface stresses, necessary to maintain the geometric

distortion of the tread surface, exceed the local frictional stresses available. The leading

edge is pointing in the steering direction while the rearward portion lags behind on the

old heading due to slip.

Figure 2-5 Contact Area Shapes at Different Slip Angles (Haney, 2004)

2.1.2.3 Tire vertical force

When a vehicle is placed on the road, it is obvious that a vertical contact force exists

between the tire and road surfaces. It is a static force due to gravity. However, when the

vehicle is running on a rough road, the tire and sprung mass have vertical accelerations.

Due to their vertical accelerations, the dynamic vertical force acts on the tire-road contact

area that can reach up to three times higher than a static vertical force. The vertical force

on the tire is affected mostly by the vertical acceleration of the sprung mass rather than


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by the vertical acceleration of the tire itself because the weight of the sprung mass is

much higher than that of the tire. The vertical forces on the tire are not applied at a point

but are distributed as normal pressure in the contact area as seen in Figure 2-6.

Fz_R

s

CompressionExtension

(a) Non-rolling stationary state (b) Rolling state

Figure 2-6 Normal Pressure Distributions in the Contact Areas (Gillespie, 1992)

Figure 2-6 (a) shows the normal contact pressure distribution in the contact area for a

non-rolling stationary tire. Because tire geometry and boundary conditions are symmetric

about the center of the contact area, the pressure distribution is also symmetric. It is noted

that at rated vertical load and inflation pressure, greater normal contact pressures are

observed under the sidewalls and centerline of the tire due to higher vertical stiffness at

those local areas. Meanwhile, Figure 2-6 (b) shows the normal contact pressure

distribution in the contact area for a rolling tire. In this case, the boundary conditions are

not symmetric about the center of the contact area. Instead, compression in the leading

portion of the tire and extension in the trailing portion of the tire near the contact area are

observed. Therefore, relatively greater normal contact pressures are exerted in the leading

area of the contact. As a result, the resultant vertical force to the tire, Fz_R, can be drawn

toward the leading edge with an offset, s, as seen in Figure 2-6 (b). This resultant force

and offset generate a moment about the tire center, O, oppositely to the tire rotational

direction, which is called the rolling resistance moment.


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2.1.2.4 Tire overturning moment

An overturning moment is defined as the moment acting on the tire spindle about the

longitudinal axis (X-axis) due to non-symmetric vertical pressure distribution across the

tire width in the contact area. Deformation of the tire casing and tread due to steering

input not only affects cornering forces but also affects the pattern of the vertical contact

force distribution in the contact area as shown in Figure 2-7.

Figure 2-7 Overturning Moment (Yap, 1991)

The magnitude and direction of the contact forces are skewed across the width of the

contact area, which results in the lateral offset slightly from the non-steered center line.

The coupling of this lateral offset and the resultant vertical force (FZ) produces an

overturning moment (MX) acting on the tire about the longitudinal axis. The overturning

moment increases linearly as slip angle and/or vertical load increase. The magnitude of

the moment is about half of that observed for self-aligning moment in the slip angle range

that is less than 4.


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2.1.2.5 Tire rolling resistance moment

When the tire is loaded and rolls on a road, the vertical contact pressure is distributed

over the contact area unevenly. Since the leading part of the tire at the contact area

undergoes compression and the trailing part of the tire undergoes extension, the vertical

resultant reaction force tends to shift toward the leading edge as shown in Figure 2-6 (b).

Due to the offset and resultant reaction force from the ground, the moment can be

developed against the tire rotational direction, which is defined as the rolling resistance

moment.

2.1.2.6 Tire vertical moment

The moment acting on the tire spindle about the vertical axis (Z-axis) is defined as the

vertical moment. Non-symmetric contact force distribution on tire-road contact plane

determines the vertical moment. In addition, during a cornering maneuver, the resultant

cornering force acts on the tire with some offset behind the center of the contact area,

called the pneumatic trail, PT, in Figure 2-4 (b). The cornering force and offset creates

the vertical moment that tends to restore the steered tire to the original unsteered tire

orientation. Therefore, this vertical moment is called the self-aligning moment or aligning

moment. The self-aligning moment increases with increasing slip angle input, similarly to

the cornering force response. The self-aligning moment increases as slip angle increases

until a peak self-aligning moment is developed at approximately 4 to 6of slip angle.

However, once the peak is reached, the self-aligning moment tends to decrease with

further increase of the slip angle because of the decrease in the moment arm, PT.

2.2 Tire Models

Much research has been conducted in the area of tire manufacture, testing, and model

development since Charles Goodyear invented the first pneumatic bicycle tires in 1839.

To accomplish laboratory tire testing, large scale experiment set-up and highly


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experienced measurement skill and effort are required. Specifically, the transient

response measurements of a tire need more complex experiment facilities and data

acquisition system. Therefore, modern scientists started to develop an alternative method

to minimize the amount of physical laboratory tire testing.

A number of tire models and virtual testing environments have been extensively

developed since early 1980s. Analytical composite material models, flexible or rigid ring

model, FEA tire models, two-dimensional models, three-dimensional models, half

models, and full models of tires have been developed and analyzed. However, tire

modeling has always been challenging because of the highly nonlinear nature of tires.

Geometric nonlinearity due to large deformations of rubber compounds, material

nonlinearity, incompressibility constraint on the deformation of rubber, and nonlinear

contact boundary conditions contribute to the difficulty of tire modeling (Oden et al.,

1982; Rother, 1984; Kenny, 1988; Tseng, 1989). Therefore, tire models that cover only

specific topics of interest have been developed.

Tire models that can predict tire responses precisely play an important role in dynamic

vehicle simulations. Allen et al.(1995) emphasized the importance of accurate tire force

modeling in dynamic vehicle simulations because these tire forces considerably affect the

dynamic vehicle handling. As shown in Figure 2-8, the tire force responses interact with

wheel spin modes, steering system, and vehicle dynamics. In other words, steering,

braking, driving torque, and vertical load play roles as inputs to the tire. Therefore, Allen

et al.concluded that all these complex interactions should be taken into account in order

to accurately model tire force responses and eventually to predict reasonable and

acceptable vehicle body motions.

As mentioned above, tire modeling is considerably challenging. Therefore, much

research is still continuing to develop more reliable and efficient tire models. Once a

good tire model is developed, it is obvious that the tire model simulation will minimize

the amount of expensive tire testing and serve for efficient dynamic vehicle simulations.


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FX: Longitudinal tire force

FY: Lateral tire force

FZ: Vertical tire force or load

MZ: Tire aligning moment

S: Tire longitudinal slip ratio

TB: Brake torque

Tp: Engine power torque

: Tire side slip angle: Tire camber angleSW: Steering wheel angleW: Front wheel angleB: Body roll angle

Figure 2-8 Tire Force Response Interactions (Allen et al., 1995)

2.2.1 Early Tire Models

Since the 1950s, string models, ring models, and beam-on-elastic foundation models

have been developed. These early tire models generally considered the tread to be a pre-

stressed string or a ring and the sidewalls to be elastic foundations supporting the tread

structure. These models mostly include a set of equivalent tire parameters that can be

determined from comprehensive tire test experiments. These analytical tire models were

useful in predicting overall tire characteristics such as vibration and responses tocornering, braking, and traction. They provided an understanding of such behavior before

more direct techniques were applied to tire analysis. However, these early models showed

limited accessibility and capability because (a) they required extensive experiments to

determine the characteristic tire parameters, (b) their validity was limited to specific

ranges of those parameters, and (c) their domain of validity could not always be predicted

in advance.

The early tire models mostly adopted the point contact mechanism because of its

simplicity (Captain et al., 1979; Loo, 1985; Loeb et al., 1990). The point contact

mechanism was established based on the assumption that a tire contacts the road surface

only through a single point which is just located under the wheel center as seen in Figure

2-9 (a). Because only the single point has contact with the road surface, the tire response


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is quite sensitive to road irregularities especially to short wavelength of road profile that

is usually filtered through a contact area in real tire applications. Therefore, the point

contact tire model is more useful for long wave road profile inputs.

To overcome this limitation of the point contact tire model, an effective road input model

and an equivalent plane tire model were established. The effective road input tire model

was developed to provide more realistic road input to the tire model as seen in Figure 2-9

(b). Due to the curvature effect of the imaginary rigid ring which contacts the road

profile, the lower frequency of the modified road profile, called the effective road profile,

could be generated as represented by the dotted line in Figure 2-9 (b). This effective road

profile provided more realistic road input to the tire than the point contact mechanism

did.

(a) Point contact mechanism (b) Effective road profile mechanism

Figure 2-9 Tire-Road Contact Mechanisms of the Early Tire Models

(Sui and Hirshey, 1999)

Meanwhile, an equivalent plane tire-road contact model was created under theassumption that the tire could be simplified as a series of linear radial springs that

connected the wheel center and the imaginary equivalent plane as seen in Figure 2-10 (b).


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(a) (b)

Figure 2-10 Equivalent Plane Tire-Road Contact Model (Sui and Hirshey, 1999)

The equivalent plane represented the road profile whose position and orientation could be

determined according to the original road profile and deformed area. Therefore, the

centroid locations of both the deformed area in Figure 2-10 (a) and the equivalent areas in

Figure 2-10 (b) are identical. In addition, the resultant force applied to the tire center due

to the spring deformations of the equivalent plane tire model was identical to the force

from the real road profile. This equivalent plane tire model could filter high frequency

road profile input and worked more precisely for concave road surfaces rather than

convex road surfaces. Still, the equivalent plane tire model has difficulty in determiningthe equivalent plane and out-of-plane behavior since the model consists of only two-

dimensional in-plane radial springs (Davis, 1974). The effective road input model

implemented a virtual road profile, which represented the wheel center trace when the

vehicle travels very slowly over the real ground surface. The point contact tire model on

this effective road profile resulted in good correlation to the real measurement output

(Guo, 1993; Guo, 1998).

In 1984, Takayama and Yamagishi introduced a lumped mass-spring tire model to

analyze the in-plane tangential and radial axial forces during cleat-drum excitation. The

tread and belts were modeled by using a rigid ring, and the tire deflection due to cleat

excitation was allowed by line or plane springs attached to the rigid ring as seen in Figure

2-11.


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Figure 2-11 Five Degrees of Freedom of Lumped Mass-Spring Tire Model

(Takayama and Yamagishi, 1984)

As shown in the Figure, the tire model consists of five degrees of freedom including

longitudinal (x1) and vertical (x2) motion of the rigid ring, local longitudinal (x3) and

vertical (x4) deflections on the cleat, and rotational motion of the rigid ring ( ). The tireaxle was fixed after loading to detect the tangential and vertical reaction force during

cleat excitation. Thus, the effects on or by the suspension system were ignored. The

displacements and forces out of the tire plane were also neglected. The parameters for the

rigid ring tire model were determined through comprehensive experiments to represent a

passenger car tire, 165SR13. The tire model was loaded at 3,780N (850 lb) on a cleat-

drum and driven at 40 km/h (25 mph) to predict longitudinal and vertical axial force and

vibration modes. The predicted axial forces agreed well with the measured forces. It was

found that the lowest frequency of vibration mode was detected at 40 Hz for the

rotational vibration mode. The predicted lowest frequencies of longitudinal vibration

mode at 67 Hz and vertical vibration mode at 74 Hz also agreed well with experimental

results.


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In 1985, Loo developed an analytical tire model which consisted of a flexible ring under

tension with a nest of radially arranged linear springs and dampers to represent a

pneumatic tire model. He was concerned with the prediction of the tires vertical load-

displacement characteristics and its free rolling resistance. The ring, which represented

the tread band of the tire, was assumed to be massless and completely flexible. The

mathematical formulation of the tire contact with the smooth hard surface was based on

the theory of a tensioned string supported by an elastic foundation. Forces developed

within the contact area were computed from a geometrical perspective. The experimental

verification was conducted and showed good correlation by comparison with the

predicted vertical load-deflection characteristics for the normal working range of inflation

pressure, deflection, and increasing surface curvature. The predicted rolling resistance

characteristics were also found to be in good accord with the experimental measurements.

Since 1987, Pacejka and his colleagues have established mathematical formulas to predict

cornering forces and aligning moments. These formulas, called the Magic Formulas, are

based on tire measurement data and calculate tire forces or moments at similar operation

conditions (Bakker et al., 1987; Bakker et al., 1989; Apetaur, 1991; Hirschberg, 1991;

Lidner, 1991; Oosten and Bakker, 1991; Pacejka and Bakker, 1991, Pacejka et al., 1997).

To complete a set of Magic Formulas, tire measurement data, such as cornering force

versus slip angle, self-aligning moment versus slip angle, or brake force versus slip ratio,

should be prepared in advance. Therefore, extensive tire measurements need to be

performed to cover a certain range of vertical tire loads. Once the Magic Formulas are

established for a certain operation condition, the calculated output correlates quite well

with the measurement data. Meanwhile, in order to generalize the Magic Formulas at a

wide range of vertical loads, 13 coefficients need to be calculated from the extensive and

costly tire measurements at various vertical tire loads. Because of the great number of the

coefficients, the Magic Formulas method is frequently criticized. The example of the

coefficients in the Magic Formula for a car tire is provided in Bakker et al.s work in

1987.


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In 1997, Zegelaar and Pacejka constructed a rigid ring tire model to represent a passenger

car tire as seen in Figure 2-12. In the rigid ring tire model, the tread and steel belts were

modeled together as a rigid ring and placed on an elastic foundation that represented the

tire sidewall. Since the tread and steel belts were modeled as a rigid ring, a new

parameter such as a vertical residual stiffness was required to describe the large

deformation of the tire in the contact area. A longitudinal slip model was also introduced

that generated the longitudinal contact force in the contact area. In order to determine the

required parameters for the rigid ring tire model, measured tire frequency response on a

rotating 2.5 m-diameter drum was used.

In-plane rotational sidewall

stiffness and damping

Tire belt

Rim

In-plane longitudinalsidewall stiffness and

damping

In-plane vertical

sidewall stiffness and

damping

Residual vertical

stiffness and dampingRoad surface

Longitudinal slip model

Figure 2-12 In-Plane Rigid Ring Tire Model (Zegelaar and Pacejka, 1997)

Zegelaar and Pacejka stated that tire in-plane vibrations are attributed to three major

sources: brake torque fluctuations, road unevenness, and horizontal and vertical

oscillations of the axle. Previously, they had presented the dynamic tire responses on


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uneven roads. In this study, longitudinal force and rotational velocity at brake pressure

variations were predicted by using the rigid ring tire model. The results from the

simulation showed good correlation with measurements. Meanwhile, from free rolling

simulations, the rigid ring tire model could successfully present an increasing effective

rolling radius and vertical contact force as the tire rolling speed increased, which was due

to the increased centrifugal forces of the tire sidewall and tread.

In 1997, Kim and Savkoor performed an analysis of the in-plane contact problem of free

rolling pneumatic tires on a flat road by using a ring tire model. The tire model was

mainly constructed with an elastic ring supported on a viscoelastic foundation as shown

in Figure 2-13.

Figure 2-13 Flexible Ring Tire Model (Kim and Savkoor, 1997)

The elastic ring represented a flexible tread band and belts of the tire while the elastic

foundation described deformable sidewalls. In addition, elastic spring components, KEt

and KGt, were introduced on the outer surface of the elastic ring to model the radial and

tangential flexibility of tire tread rubber. The input parameters required to complete the

ring tire model are listed in Figure 2-13. They determined those parameters based on

measurements such as vertical load versus deflection of a radial passenger car tire,


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205/60R15. Then, traction force distributions at contact area and rolling resistance

coefficients were predicted at different speeds. However, no experimental measurements

were provided to validate their predicted tractive contact forces and rolling resistance

coefficients from the ring model.

Bruni et al.in 1997 proposed a methodology to determine the in-plane tire parameters for

a rigid ring tire model from limited experimental tests. The rigid ring tire model was

devised such that it could perform vehicle comfort, braking, and driving analysis as seen

in Figure 2-14.

Figure 2-14 Rigid Ring Tire Model (Bruni et al., 1997)

To complete the rigid ring tire model, basic parameters such as the moment of inertia of

the rigid ring, rigid ring mass, and effective rolling radius were directly measured through

torsional pendulum and free rolling tests. Then, most of the rest of the parameters, such

as sidewall stiffness and damping as well as residual stiffness and damping were

estimated by minimizing the difference between the experimental and analytical damped

natural frequencies. The required in-plane tire parameters were determined for a


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passenger car tire, 185/60R14, at inflation pressure of 0.20 MPa (29 psi) and vertical load

of 1,990 N (450 lb) and validated through natural frequency and damping factor tests.

Allison and Sharp (1997) also examined this simple rigid ring model to approach the low

frequency (up to about 100 Hz) in-plane longitudinal vibration problems of vehicles.

Schmeitz et al. (2004) presented a quarter vehicle model by combining the rigid ring tire

model, suspension, sprung mass and elliptical cams together. The elliptical cams were

adopted to generate an effective road profile. They predicted vertical tire motions and

longitudinal forces for different heights of step road inputs. Then, the predicted results

were compared with measurements and showed good correlations. They also conducted

modal analyses on the quarter vehicle system to find a vertical mode at 71.5 Hz and a

horizontal mode at 84.4 Hz.

2.2.2 Finite Element Analysis Tire models

The finite element analysis (FEA) is a useful numerical tool especially in evaluating a

complicated design model. Since 1970, FEA has been widely adopted in analyzing stress,

strain, and elastic/plastic deformation of more complicated structural models which

cannot be accomplished with ease by experimental measurements or other simple models.For example, structural models in very high pressure or temperature application, vehicle

crashworthiness, and high speed impact simulations of deformable bodies have been

constructed and analyzed by using FEA.

Since 1970s, FEA has been adopted in the development of effective and efficient tire

models because traditional structural analysis techniques could no longer offer

sufficiently detailed results for advanced tire designs. Moreover, the FEA tire models can

reflect real-world boundary conditions more directly without adjustments compared to

the previous non-FEA tire models. The FEA tire models can also directly handle large

deformations around the contact area without additional adjustments such as the residual

stiffness and damping used in ring tire models. The FEA tire models can also use an

actual road profile, without any modification of the original profile. Early FEA tire


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models can predict basic characteristic tire responses, such as tire-road contact areas,

stiffness, vibration modes, cornering forces, and self-aligning moments.

The FEA simulation of a tire is still a challenging task due to computational contact

problems as well as the nonlinearity nature of the tire. The handling of the contact

boundary is the most challenging and has attracted much intention. In many cases, a

Lagrange multiplier, penalty term, or gap element is adopted to describe the interaction

between tire treads and the road surface. A gap element is the easiest way to handle

contact problems. However, it cannot easily deal with contact-impact and contact on an

arbitrarily shaped surface. In this section, the FEA tire models with contact are reviewed.

The development of the contact solution algorithms are reviewed in Section 2.3. Due to

computational limitations, most tire simulations using the finite element method have

been limited to the static or quasi-static analyses in early ages of FEA tire models. It was

not until in late 1990s that transient responses have been studied using an explicit FEA

option.

Padovan (1977) employed a two-dimensional curved axis-symmetric thin shell element

model to examine the power dissipative rolling resistance and thermo-viscoelastic

problem of steady-state rolling tires. Trivisonno (1977) also examined the similar thermal

problem by a two-dimensional finite difference model. Noor and Anderson (1982) stated

that thin shell models neglect transv

Seokyong Chae PhD Thesis ID 993173980

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